What is the integral of (x+1)(x^2+2x+5)^{12} ?

The integral of (x+1)(x^2+2x+5)^{12} is 1/26 (x^2+2x+5)^{13}+C

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integral of (x+1)(x^2+2x+5)^{12}

\int (x + 1) (x^{2} + 2 x + 5)^{12} d x

\frac{1}{26} (x^{2} + 2 x + 5)^{13} + C

Solution steps

\int (x + 1) (x^{2} + 2 x + 5)^{12} d x

Apply u-substitution

: \int u (u^{2} + 4)^{12} d u

= \int u (u^{2} + 4)^{12} d u

Apply u-substitution

: \int \frac{v ^{12}}{2} d v

= \int \frac{v ^{12}}{2} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int v^{12} d v

Apply the Power Rule

: \frac{v ^{13}}{13}

= \frac{1}{2} \cdot \frac{v ^{13}}{13}

Substitute back

= \frac{1}{2} \cdot \frac{( ( x + 1 ) ^{2} + 4 ) ^{13}}{13}

Simplify

\frac{1}{2} \cdot \frac{( ( x + 1 ) ^{2} + 4 ) ^{13}}{13} : \frac{1}{26} (x^{2} + 2 x + 5)^{13}

= \frac{1}{26} (x^{2} + 2 x + 5)^{13}

Add a constant to the solution

= \frac{1}{26} (x^{2} + 2 x + 5)^{13} + C

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What is the integral of (x+1)(x^2+2x+5)^{12} ?
The integral of (x+1)(x^2+2x+5)^{12} is 1/26 (x^2+2x+5)^{13}+C